上海交通大学：《热力学 Thermodynamics（I）》课程教学资源（课件讲义）Lecture 14_cv, cp, Δu, Δh of ideal gas and applied to close system

上游充通大￥ SHANGHAI JIAO TONG UNIVERSITY Engineering Thermodynamics I Lectures 14 Spring,3/19/2019 Prof.,Dr.Yonghua HUANG 强 MMALLLAAMAR http://cc.sjtu.edu.cn/G2S/site/thermo.html 1日G

Engineering Thermodynamics I Lectures 14 Spring, 3/19/2019 Prof., Dr. Yonghua HUANG http://cc.sjtu.edu.cn/G2S/site/thermo.html

△u,△h,cy,and cp relations for ideal gas ideal gas:uAT);hfT) kJ/kg.K →c,definition: du c,(T)= (ideal gas) 0 du=c,(T)dT u()-()c(TdT (ideal gas) →C,definition: c,(T)=d dh dT (ideal gas) The key is to ↓ find c,(T dh=c,(T)dT and cp(T) h()h()e.(TdT Adeal sas) 上游充通大学 March 19,2019 2 SHANGHAI JLAO TONG UNIVERSITY

March 19, 2019 2 Δu, Δh, cv , and cp relations for ideal gas ideal gas: u~f(T); h~f(T) cv definition:  cp definition: d ( ) (ideal gas) d v u c T T  ( )d v du c T T  d ( ) (ideal gas) d p h c T T  d ( )d p h c T T  2 1 2 1 ( ) ( ) ( )d (ideal gas) T v T u T u T c T T    2 1 2 1 ( ) ( ) ( )d (ideal gas) T p T h T h T c T T    The key is to find cv(T) and cp(T) kJ/kg·K

cy Cp and Gas Constant Ideal gas:h u +pv =u +RT differentiating +R 'dT Mayer's formula c(T)=c(T)+R (ideal gas) On molar basis cp(T)=c(T)+R (ideal gas) 上游究通大学 March 19,2019 3 SHANGHAI JLAO TONG UNIVERSITY

March 19, 2019 3 cv , cp and Gas Constant Ideal gas: h = u +pv =u +RT d d d d h u R T T   cp cv ( ) ( ) (ideal gas) p v c T c T R   On molar basis c T c T R p v ( ) ( ) (ideal gas)   p v p v c c c c   Mayer’s formula differentiating

Explain ⊙a"→b q,=△uab+Wab +1 ap→c qp=△lac+Wac =△uac+p(ye-va) 2=1=+0K-=a T=T=T y.>y。→p(.-a)>0→9n>g And qp Cp(T:-Ta)=Cp(T+1-T)=cp C -Cp> 4,=C,(T6-Ta)=c,(T+1-T)=c 上游充通大 March 19,2019 4 SHANGHAI JLAO TONG UNIVERSITY

March 19, 2019 4 v v ab ab a b q u w     0   p p ac ac ac c a a c q u w u p v v           0 c a c a p v v v p v v q q       And         1 1 p p c a p p v v b a v v q c T T c T T c q c T T c T T c             v p Explain ( )1 K ab a a b c a c T T u T T u T T             p v c c 

Question: Is c,always greater than c,at the same temperature for any phase of any substance 上游充通大学 March 19,2019 5 SHANGHAI JLAO TONG UNIVERSITY

March 19, 2019 5 Is cp always greater than cv at the same temperature ? Question: for any phase of any substance